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This paper addresses a fundamental theoretical problem in mathematical physics by developing obstruction theory for linearizing quantum cellular automata (QCA) representations using algebraic K-theory. By computing homotopy types of QCA spaces, the authors establish universal classes that determine when these localized quantum systems can be represented linearly over arbitrary fields. The work advances the rigorous understanding of locality constraints and representation structures within quantum information theory and operator algebras.
$K$-Theoretic Obstructions to Linearizing QCA Representations
arXiv:2606.19657v2 Announce Type: replace-cross Abstract: Projective representations arise naturally in physics and representation theory, and determining whether they can be linearized has been a fundamental problem. In this work, we study the analogous problem for quantum cellular automata (QCA) representations, which incorporate locality constraints imposed by a metric space $X$. Over an arbitrary field $\mathbb{F}$, we develop an obstruction theory for the linearization of QCA representations, using the algebraic $K$-theory spectrum of QCA constructed in previous work of the authors. The resulting obstructions are governed by the homotopy type of the QCA spaces, from which we extract universal obstruction classes to linearization. In the complex algebraic and unitary case, we also fully compute the homotopy types of the QCA spaces over a point, a line, and a plane.
Abstract:Projective representations arise naturally in physics and representation theory, and determining whether they can be linearized has been a fundamental problem. In this work, we study the analogous problem for quantum cellular automata (QCA) representations, which incorporate locality constraints imposed by a metric space $X$. Over an arbitrary field $\mathbb{F}$, we develop an obstruction theory for the linearization of QCA representations, using the algebraic $K$-theory spectrum of QCA constructed in previous work of the authors. The resulting obstructions are governed by the homotopy type of the QCA spaces, from which we extract universal obstruction classes to linearization. In the complex algebraic and unitary case, we also fully compute the homotopy types of the QCA spaces over a point, a line, and a plane.
Algebraic Topology (math.AT); Mathematical Physics (math-ph); Operator Algebras (math.OA); Representation Theory (math.RT); Quantum Physics (quant-ph)
arXiv:2606.19657 [math.AT]
(or arXiv:2606.19657v2 [math.AT] for this version)
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